Lecture 12: Integral representation of the basic
hypergeometric function q -Integration by parts.
Aims:
Finish our discussion of q -integrals
Prove the q -integral representation of the basic hypergeometric function.
Demonstrate q -integration by parts
Lecture 19: More q -Ladder operators.
Aims:
To nd the linearly independent solution to the q -dierence equation in
terms of the associated function.
19.1. q -dierence-dierence equations for orthogonal polynomials.
We now have a system of q -dierence equa
Lecture 20: Example:q -Hermite Polynomials
Aims:
To get a grip on the mechanics of the developed theory by detailing an
example.
Last lecture we showed that the discrete Hermite polynomials, dened by
1
(1 qx) (1 + qx) f (x)g (x)dq x.
q
q
f, g =
1
and the
Lecture 23: Singularity connement.
Aims:
Dene singularity connement for a mapping
Let us consider the QRT mapping arising from the integral
In =
1
1
1
+
+ xn1 + xn +
xn
xn1
xn xn1
so we calculate, In+1 In with
1
1
1
xn+1
x n1
xn xn+1
x n1
xn1 xn+1
xn1 xn
Lecture 21: q -Airy and q -Bessel functions.
Aims:
To introduce the q -analogues of the Airy and Bessel functions
20.1. q -Airy functions. Let us start with the Airy functions. The Airy function is a solution of the dierential equation
d2 y
= xy,
dx2
(20
Lecture 15:More denitions and examples.
Aims:
To discuss the three term recurrence relation further for orthogonal polynomils.
Present more examples that motivate the presented theory.
Last lecture, an important point was the denition of the three term
Lecture 18: q -Ladder operators.
Aims:
To determine the q -dierence equation satised by orthogonal polynomials.
18.1. q -dierence equations for orthogonal polynomials. The aim of
this section is to nd a q -dierence equation satised by an orthogonal polyn
Lecture 14: Orthogonal polynomials.
Aims:
Give some sort of preface to the courses treatment of orthogonal polynomials
Describe some of the common elements to orthogonal polynomials and
q -orthogonal polynomials.
Orthogonal polynomials are steeped in hi
Lecture 17: More ladder operators.
Aims:
To complete our description of the linear problem for orthogonal polynomial ensembles
Provide the linearly independent
Furnish our work with a simple example.
Last lecture we proved that
d
pn (x) = (n (x) V (x)p
Outline of the course
I Discrete Calculus :
q -Derivatives and h-Derivatives.
Discrete product rules and quotient rules.
Discrete Taylor expansions.
The two q -exponential functions.
q -Trigonometric functions.
II q -Special Functions:
q -Integratio
Lecture 2: Taylors Formula.
Aims:
Present a generalized form of Taylors Theorem.
2.1. Taylors Theorem. Let us consider a function, f , which possesses all
its derivatives at a point a. Such a function is called analytic at x = a. We may
reconstruct f (x)
Lecture 5: Binomial identities.
Aims:
To give some analogous q -binomial identities.
To present a combinatorial application of the q -calculus.
5.1. Binomial Identities. Last lecture we proved two versions of q -Pascals
rule:
Proposition 5.1. The follow
Lecture 11: The q -integral.
Aims:
Discuss the general q -antiderivative/integral.
Introduce Jacksons denite q -integral.
11.1. Antiderivative. Some functions are naturally expressed in terms of
integrals, such as the Gamma function, the Beta function.
Lecture 9: q -hypergeometric functions.
Aims:
Review hypergeometric functions.
Introduce generalized basic hypergeometric functions.
9.1. Gausss hypergeometric function. Inevitably, any discussion of special functions should make a proper mention to hyp
Lecture 13: q -Gamma q -Beta functions.
Aims:
To describe two q -functions, the q -Gamma and q -Beta functions, in terms
of integrals
To describe the q -logarithm.
13.1. Gamma and Beta functions. The Gamma function is dened by the
denite integral
(13.1)
Lecture 10: Sums and transformation formulas
Aims:
Exhibit some basic hypergeometric transformations.
Demonstrate a summation formula.
10.1. Heines Transformation formula. A basic hypergeometric representation for a given function is by no means unique.
Lecture 7: q -exponential functions.
Aims:
Consider the innite products (1 + x) .
q
Introduce the two analogues of the q -exponential function.
7.1. The q -exponential functions and (1 + x) . We wish to come to a
q
solution to an important q -dierence e
Lecture 3: Taylors formula continued.
Aims:
To continue our discussion of Taylor polynomials.
Demonstrate some basic identities.
3.1. The q -Taylor polynomials. So last lesson, the polynomials specied
by our generalized Taylors theorem for a = 0 were gi
Lecture 4: q -Binomials.
Aims:
To introduce the q -Binomials
Introduce some (non-assessable) quantum systems.
4.1. q -Binomials. We have shown that the sequence of polynomials, given by
Pn (x) =
(x a)n
q
[n]!
satises the requirements of theorem 2.1 from
Lecture 6: q -Fibonacci Numbers
Aims:
Present the q -Fibonacci numbers.
Apply the q -Pascals rule.
Explore extensions of generating functions for q -dierence equations.
6.1. q -Fibonacci numbers. We work so hard sometimes its nice to goof o
and do some
Lecture 16: Ladder operators.
Aims:
Itroduce the determine a dierential equation satised by an orthogonal
polynomial ensemble.
A topic that is close to my heart (I have a fair few results in this direction)
is the topic of Ladder operators. We are lookin